The industry is at it again–trying to figure out what to make of Metcalfe’s Law. This time it’s IEEE Spectrum with a controversially titled “Metcalfe’s Law is Wrong”. The main thrust of the argument is that the value of a network grows O(nlogn) as opposed to O(n2). Unfortunately, the authors’ O(nlogn) suggeston is no more accurate or insightful than the original proposal.

There are three issues to consider:

The difference between what Bob Metcalfe claimed and what ended up becoming Metcalfe’s Law

The units of measurement

What happens with large networks

The typical statement of the law is “the value of a network increases proportionately with the square of the number of its users.” That’s what you’ll find at the Wikipedia link above. It happens to not be what Bob Metcalfe claimed in the first place. These days I work with Bob at Polaris Venture Partners. I have seen a copy of the original (circa 1980) transparency that Bob created to communicate his idea. IEEE Spectrum has a good reproduction, shown here.

The unit of measurement along the X-axis is “compatibly communicating devices”, not users. The credit for the “users” formulation goes to George Gilder who wrote about Metcalfe’s Law in Forbes ASAP on September 13, 1993. However, Gilder’s article talks about machines and not users. Anyway, both the “users” and “machines” formulations miss the subtlety imposed by the “compatibly communicating” qualifier, which is the key to understanding the concept.

Bob, who invented Ethernet, was addressing small LANs where machines are visible to one another and share services such as discovery, email, etc. He recalls that his goal was to have companies install networks with at least three nodes. Now, that’s a far cry from the Internet, which is huge, where most machines cannot see one another and/or have nothing to communicate about… So, if you’re talking about a smallish network where indeed nodes are “compatibly communicating”, I’d argue that the original suggestion holds pretty well.

The authors of the IEEE article take the “users” formulation and suggest that the value of a network should grow on the order of O(nlogn) as opposed to O(n2). Are they correct? It depends. Is their proposal a meaningful improvement on the original idea? No.

To justify the logn factor, the authors apply Zipf’s Law to large networks. Again, the issue I have is with the unit of measurement. Zipf’s Law applies to homogeneous populations (the original research was on natural language). You can apply it to books, movies and songs. It’s meaningless to apply it to the population of books, movies and songs put together or, for that matter, to the Internet, which is perhaps the most heterogeneous collection of nodes, people, communities, interests, etc. one can point to. For the same reason, you cannot apply it to MySpace, which is a group of sub-communities hosted on the same online community infrastructure (OCI), or to the Cingular / AT&T Wireless merger.

The main point of Metcalfe’s Law is that the value of networks exhibits super-linear growth. If you measure the size of networks in users, the value definitely does not grow O(n2) but I’m not sure O(nlogn) is a significantly better approximation, especially for large networks. A better approximation of value would be something along the lines of O(SumC(O(mclogmc))), where C is the set of homogeneous sub-networks/communities and mc is the size of the particular sub-community/network. Since the same user can be a member of multiple social networks, and since |C| is a function of N (there are more communities in larger networks), it’s not clear what the total value will end up being. That’s a Long Tail argument if you want one…

Very large networks pose a further problem. Size introduces friction and complicates connectivity, discovery, identity management, trust provisioning, etc. Does this mean that at some point the value of a network starts going down (as another good illustration from the IEEE article shows)? It depends on infrastructure. Clients and servers play different roles in networks. (For more on this in the context of Metcalfe’s Law, see Integration is the Killer App, an article I wrote for XML Journal in 2003, having spent less time thinking about the problem ;-)). P2P sharing, search engines and portals, anti-spam tools and federated identity management schemes are just but a few examples of the myriad of technologies that have all come about to address scaling problems on the Internet. MySpace and LinkedIn have very different rules of engagement and policing schemes. These communities will grow and increase in value very differently. That’s another argument for the value of a network aggregating across a myriad of sub-networks.

Bottom line, the article attacks Metcalfe’s Law but fails to propose a meaningful alternative.

26 Responses to Metcalfe’s Law: more misunderstood than wrong?

hm, a small community needs to have a high proportion of links in order to be “small” in terms of Metcalfe’s law (e.g. a community of three needs mutual links between all three of its nodes); whereas the value of the proportion between the number of links and the number of nodes decreases as the community becomes bigger: you don’t expect a community with a thousand nodes to have all of these nodes link to each other in order to call it “small” (in proportion). So this sounds like another use for a Long Tail graph: very few big communities will bother to establish an unusually high number of links, and most will go along with the mean values in the long tail of the graph.

What I do not understand about all of this is how one can argue over the value of something which is largely subjective. Certainly, the *potential* value of a network can increase with unit size, but it also has can decrease over time as well. The classic fax machine example also demonstrates this.

These ‘Laws’ are theoretical. Since there is no empirical manner in which to calculate value, each ‘Law’ is subjective and lacks evidence to support them. Because of the nature of the ‘Law’, it can never be proven – definitely convenient, and poor science.

It is intuitive that the value of a network would increase as the size of the network increases, but assessing it in an equation seems a bit like hubris to me.

One application goes to economics. In a supply chain you may have a web of suppliers creating concurrence. The value depends on each company but assuming it constant then the value increases and the availability of souces increase.

The point of the value of a network changing over time is interesting. The value of a Fax machine is changing, but in some ways it is actually continuing to grow as its functionality as a scanning device continues to be utilized. Also. Its value in times of disaster (which we seem to be seeing increasingly having) goes up even more. During a disaster it is a standard backup machine not dependent on the Internet infrastructure, that can retry and provide confirmation of receipt. The Fax machine is continues to be used as a copy machine or backup copy machine in growing use (although more by the ever increasing small home worker). However, its original use of point-to-point communication is decreasing in the ‘mainstream’ world with email, it is still used for distribution, and for accessing “lower end users” without all the higher dollar technology solutions.

The value of a network is equal to the number of links in the network. For a *fully connected* network, where every node has a direct connection to every other node, that number matches melcalfe’s law i.e. o(n squared).

But for e.g. social networks, the number of links is really more o(n). This is because the number of other user’s that the average user is directly connected to is not a function of the overall size of the network.

If there are a 100,000 users in a network, and the average number of connections is 10 (say), then the value of the network is (1000 * 10 ) /2 = 500,000 – and not 10,000,000,000.

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